Questions C3 (1200 questions)

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Edexcel C3 Q7
7. \(\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the range of f.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
  5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).
Edexcel C3 Q8
8. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    [0pt]
  2. Show that the curve has a stationary point in the interval [1.3,1.4]. The point \(A\) on the curve has \(x\)-coordinate 2 .
  3. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
    The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
  4. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.
Edexcel C3 Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8b85e00-4549-4219-a75d-85f67ccb8e16-2_638_675_644_445} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.
  1. Find the exact length \(A B\). The two curves intersect at the point \(C\).
  2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).
Edexcel C3 Q3
3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$
  1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
Edexcel C3 Q4
4. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).
  1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
  2. Find the \(x\)-coordinates of the stationary points of \(C\).
Edexcel C3 Q5
5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } ,
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$
  1. Evaluate \(\operatorname { gf } ( 1 )\).
  2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
  3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
Edexcel C3 Q6
  1. (a) Use the identities for \(\cos ( A + B )\) and \(\cos ( A - B )\) to prove that
$$\cos P - \cos Q \equiv - 2 \sin \frac { P + Q } { 2 } \sin \frac { P - Q } { 2 }$$ (b) Hence find all solutions in the interval \(0 \leq x < 180\) to the equation $$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$ Turn over
Edexcel C3 Q7
7. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$ where \(a\) is a positive constant.
  1. Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
    1. \(\quad y = | \mathrm { f } ( x ) |\),
    2. \(y = \mathrm { f } ( | x | )\). The function g is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
  2. Find fg(a) in terms of \(a\).
  3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$
Edexcel C3 Q8
8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
  3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
  4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.
Edexcel C3 Q1
  1. (a) Given that \(\cos x = \sqrt { 3 } - 1\), find the value of \(\cos 2 x\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
    (b) Given that
$$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$ find the value of \(\tan y\) in the form \(k \sqrt { 3 }\) where \(k\) is a rational constant.
Edexcel C3 Q2
2. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv x ^ { 2 } - 3 x + 7 , \quad x \in \mathbb { R }
& \mathrm {~g} ( x ) \equiv 2 x - 1 , \quad x \in \mathbb { R } \end{aligned}$$
  1. Find the range of f .
  2. Evaluate \(\operatorname { gf } ( - 1 )\).
  3. Solve the equation $$\mathrm { fg } ( x ) = 17$$
Edexcel C3 Q3
  1. \(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
    1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that
    $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$ The point \(P\) on the curve \(y = \mathrm { f } ( x )\) has \(x\)-coordinate 1.
  2. Show that the normal to the curve \(y = \mathrm { f } ( x )\) at \(P\) has the equation $$x + 5 y + 9 = 0$$
Edexcel C3 Q4
  1. (a) Given that
$$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$ (b) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).
Edexcel C3 Q5
5. $$\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , \quad x \in \mathbb { R } .$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 7\).
  4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).
Edexcel C3 Q6
6. (a) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z } .$$ (b) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7 ,$$ giving your answers to 2 decimal places.
Edexcel C3 Q7
7. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$
  1. State the range of f .
  2. Evaluate fg(-2).
  3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
  4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
  5. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
  6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.
Edexcel C3 Q1
  1. (a) Find the exact value of \(x\) such that
$$3 \arctan ( x - 2 ) + \pi = 0$$ (b) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).
Edexcel C3 Q2
2. (a) Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
(b) Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 } .$$
Edexcel C3 Q3
  1. Differentiate each of the following with respect to \(x\) and simplify your answers.
    1. \(\cot x ^ { 2 }\)
    2. \(x ^ { 2 } \mathrm { e } ^ { - x }\)
    3. \(\frac { \sin x } { 3 + 2 \cos x }\)
    4. (a) Find, as natural logarithms, the solutions of the equation
    $$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$
  2. Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.
Edexcel C3 Q5
5. The function f is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R } .$$
  1. State the range of f.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R } .$$ Find, in terms of e,
  3. the value of \(\mathrm { gf } ( \ln 2 )\),
  4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4 .$$
Edexcel C3 Q6
6. $$f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2 .$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = 2 - \mathrm { e } ^ { k x ^ { 2 } } ,$$ where \(k\) is a constant to be found. The root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) is 1.9 correct to 1 decimal place.
  2. Use the iteration formula $$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } } ,$$ with \(x _ { 0 } = 1.9\) and your value of \(k\), to find \(\alpha\) to 3 decimal places and justify the accuracy of your answer.
  3. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\).
Edexcel C3 Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7c3dd501-0545-4166-aaf9-5e1ac1f369c5-4_552_771_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ( | x | )\),
    2. \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ } ,$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
Edexcel C3 Q1
  1. \(f ( x ) \equiv \frac { 2 x - 3 } { x - 2 } , \quad x \in \mathbb { R } , \quad x > 2\).
    1. Find the range of f .
    2. Show that \(\operatorname { ff } ( x ) = x\) for all \(x > 2\).
    3. Hence, write down an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    4. Solve each equation, giving your answers in exact form.
    5. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
    6. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
    7. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.
    8. Find an equation for the tangent to \(C\) at \(P\).
    The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  2. Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).
Edexcel C3 Q4
4. (a) Express $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form.
(b) Hence, show that the equation $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) } = 1$$ has no real roots.
Edexcel C3 Q5
5. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4$$ giving your answers to 1 decimal place.